You are correct in that the full Taylor expansion of an polynomial always returns to you the original polynomial. Notice that your degree two Taylor polynomial is also your original polynomial, but without the terms with degree higher than 2.

The idea of Taylor polynomials is that any smooth function can be approximated arbitrarily well with just polynomials. In my opinion, this is a truly beautiful fact. Why should we expect this? Are there other functions we can use instead of polynomials? Maybe sines and cosines?

As for their practical uses:
As you said, they describe the behaviour of the function around a point. They show up in applied mathematics and physics a fair bit when people want to make approximations to functions. They show up in numerical mathematics, again for approximation purposes. Power series are used to solve differential equations, and the Taylor series is a special kind of power series which converges to a known function. Combinatorial enumeration uses power series, and sometimes Taylor series show up there (though only formally). The Taylor series of real functions is used to extend the idea of evaluating functions on operators; for example, we use the Taylor series expansion of the exponential to define the matrix exponential, matrix logarithm, etc.

There are classes of functions which are infinitely differentiable, but are not analytic (that is, their Taylor series do not always converge in a neighbourhood of their domain). Hence some people study analytic functions themselves: this lends itself to the difference between, say, smooth and analytic manifolds. You will see in complex analysis though that every differentiable function is in fact holomorphic! (another word that means analytic).

Taylor's theorem is actually a special case of a far more powerful theorem called the Stone-Weierstrass Theorem. If X is a compact Hausdorff space and [itex] B \subseteq C(X,\mathbb R)[/itex] (continuous functions from X to [itex] \mathbb R [/itex]) is a unital subalgebra which separates points, the B is dense in [itex] C(X,\mathbb R) [/itex].